﻿Einstein and Grossmanns Theory of Gravitation. 83 



what the acceleration will be if, availing ourselves of the 

 relation a 2 — b 2 = l } we notice from (a) that 



and, therefore, for a point with a fixed value of z, 



lt;_ J?* 



so that the velocity of the moving system will never exceed 

 the velocity of light. From this follows for the acceleration 



_cn_ /(;-co) 2 _ 



9 ~-dr*~ V^-r ) 2 + c 2 T 2 ' 3 ' 



This gives for t = the starting acceleration g as found 

 above. The constant k has no particular meaning. It re- 

 lates the value of the velocity of light as measured in the 

 moving system to the velocity of light in the other. If g 00 

 be the acceleration of the origin £ = at the time £ = 0, and 

 we want our system of coordinates defined in such a manner 

 that at this same point and time the velocity of light c' 00 be 

 the same as in the resting system, then we have to take 



k= — . For we see easily that z ~ ? and if c f o= ~~^o 



c <7oo 



is to be equal to c, then we must have k=g 00 /c, as stated. 

 Lastly, we may notice that to the differential equations 



d% — a dz + bc'dt, c dr = ac'dt + h dz 



correspond, by virtue of a 2 — ?> 2 =1, the reciprocal equations 



dz = a d£— be ar, c'dt = ac dr — b d£. 



Motion of a free particle. 



7. Now, knowing the motion of a free point through a 

 space without gravitation (and such is the motion in our 

 resting system) to be in a straight line, Einstein, by using 

 the relation of the coordinates of the two systems, could find 

 the equations of motion when referred to the moving axes. 

 He found it possible * to contract them into the form of 

 Hamilton's principle : 



S\§R'dt\=0, 



* Ann. d. Phys. xxxviii. p. 453 (1912). 



G 2 



